Accurate description of the mechanical behavior of crystalline Na alloys is essential in establishing their electrochemical performance as well as their viability as anodes in Na-ion batteries. Using first principles simulations, we have investigated the intrinsic elastic properties of crystalline Na-M (M = Sn, Pb, Si and Ge) phases observed during Na intercalation. We have obtained the complete set of concentration-dependent anisotropic elastic constants as well as the average macroscopic elastic moduli of polycrystalline structures. We find that sodiation of pure M phases leads to a remarkable elastic softening that results in up to 75% deterioration of the elastic moduli. Our analysis of the electronic charge distribution demonstrates that the elastic softening during sodiation originates from a transition to weaker ionic interatomic bonding. Our results highlight the significance of the concentration dependence of the elastic moduli for the analysis of deformation behavior of Na alloy anodes of Na-ion batteries during sodiation and desodiation. (C) 2012 Elsevier B.V. All rights reserved.